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%I #19 Sep 22 2023 12:17:23
%S 0,0,0,0,0,1,1,1,2,2,2,2,3,3,4,3,4,3,4,4,4,3,5,4,4,4,5,4,4,2,4,4,4,2,
%T 3,2,3,2,1,2,2,3,3,3,4,5,3,2,5,6,5,5,6,5,7,9,6,7,9,9,8,10,8,8,10,7,8,
%U 10,6,9,8,6,5,8,4,7,4,4,8,7,5,3,5,3,7,3,3,5,7,5,4,6,5,6,7,5,6,10,9,6
%N Number of ways to write n = a + b + c (0 < a <= b <= c) such that all the three numbers a*(a+1)-1, b*(b+1)-1, c*(c+1)-1 are Sophie Germain primes.
%C Conjecture: a(n) > 0 for all n > 5.
%C Conjecture verified for n up to 10^9. - _Mauro Fiorentini_, Sep 22 2023
%C This implies that there are infinitely many Sophie Germain primes of the form x^2 + x - 1.
%C See also A230516 for a similar conjecture.
%H Zhi-Wei Sun, <a href="/A230514/b230514.txt">Table of n, a(n) for n = 1..5000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588 [math.NT], 2012-2017.
%e a(10) = 2 since 10 = 2 + 2 + 6 = 2 + 3 + 5, and 2*3 - 1 = 5, 6*7 - 1 = 41, 3*4 - 1 = 11, 5*6 - 1 = 29 are all Sophie Germain primes.
%e a(39) = 1 since 39 = 9 + 15 + 15, and both 9*10 - 1 = 89 and 15*16 - 1 = 239 are Sophie Germain primes.
%t pp[n_]:=PrimeQ[n(n+1)-1]&&PrimeQ[2n(n+1)-1]
%t a[n_]:=Sum[If[pp[i]&&pp[j]&&pp[n-i-j],1,0],{i,1,n/3},{j,i,(n-i)/2}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A005384, A209253, A227923, A230040, A230515, A230516.
%K nonn
%O 1,9
%A _Zhi-Wei Sun_, Oct 21 2013
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